Theory of high-harmonic generation by low-frequency laser fields

Quantum path selection in high-harmonic generation by a phase-locked two-color field

We report the results of our studies on the selection of the quantum path in high-harmonic generation (HHG) with a relative-phase-locked two-color laser field. It is shown that by tuning the relative phase between fundamental and second-harmonic fields, The timing of tunnel ionization and subsequent electron trajectories on the sub-cycle time scale can be controlled. We have clearly observed a phase-dependent two-step feature in the harmonic spectra that can be attributed to the selection of two major trajectories in the two-color field HHG.

Quasi-phase-matched high harmonic generation in hollow core photonic crystal fibers

The potential of hollow core photonic crystal fiber as a nonlinear gas cell for efficient high harmonic generation is discussed. The feasibility of phase-matching this process by modulating the phase of ionization electrons using a counter-propagating laser field is shown. In this way, harmonics with energies of several hundreds of eV can be produced using fs-laser pump pulses of microJ energy.

Laser applications and system considerations in ocular imaging

We review laser applications for primarily in vivo ocular imaging techniques, describing their constraints based on biological tissue properties, safety, and the performance of the imaging system. We discuss the need for cost effective sources with practical wavelength tuning capabilities for spectral studies. Techniques to probe the pathological changes of layers beneath the highly scattering retina and diagnose the onset of various eye diseases are described. The recent development of several optical coherence tomography based systems for functional ocular imaging is reviewed, as well as linear and nonlinear ocular imaging techniques performed with ultrafast lasers, emphasizing recent source developments and methods to enhance imaging contrast.

Coherent manipulations of atoms using laser light

The internal structure of a particle - an atom or other quantum system in which the excitation energies are discrete - undergoes change when exposed to pulses of near-resonant laser light. This tutorial review presents basic concepts of quantum states, of laser radiation and of the Hilbert-space statevector that provides the theoretical portrait of probability amplitudes - the tools for quantifying quantum properties not only of individual atoms and molecules but also of artificial atoms and other quantum systems. It discusses the equations of motion that describe the laser-induced changes (coherent excitation), and gives examples of laser-pulse effects, with particular emphasis on two-state and three-state adiabatic time evolution within the rotating-wave approximation. It provides pictorial descriptions of excitation based on the Bloch equations that allow visualization of two-state excitation as motion of a three-dimensional vector (the Bloch vector). Other visualization techniques allow portrayal of more elaborate systems, particularly the Hilbert-space motion of adiabatic states subject to various pulse sequences. Various more general multilevel systems receive treatment that includes degeneracies, chains and loop linkages. The concluding sections discuss techniques for creating arbitrary pre-assigned quantum states, for manipulating them into alternative coherent superpositions and for analyzing an unknown superposition. Appendices review some basic mathematical concepts and provide further details of the theoretical formalism, including photons, pulse propagation, statistical averages, analytic solutions to the equations of motion, exact solutions of periodic Hamiltonians, and population-trapping "dark" states.

Efficient output coupling of intracavity high-harmonic generation

We demonstrate a novel technique for coupling extreme-ultraviolet (XUV) harmonic radiation out of a femtosecond enhancement cavity. We use a small-period diffraction grating etched directly into the surface of a dielectric mirror. For the fundamental light, this element acts as a high reflector. For harmonic wavelengths, it acts as a diffraction grating, coupling XUV radiation out of the cavity. Using this method, we observed the third through twenty-first odd harmonics with a dramatic increase in usable power over previous results of high-harmonic generation at high repetition rates.

Wavelength scaling of high-harmonic yield: threshold phenomena and bound state symmetry dependence

Describing harmonic generation (HG) in terms of a system's complex quasienergy, the harmonic power P_{DeltaE}(lambda) (over a fixed interval, DeltaE, of harmonic energies) is shown to reproduce the wavelength scaling predicted recently by two groups of authors based on solutions of the time-dependent Schrödinger equation: P_{DeltaE}(lambda) approximately lambda;{-x}, where x approximately 5-6. Oscillations of P_{DeltaE}(lambda) on a fine lambda scale are then shown to have a quantum origin, involving threshold phenomena within a system of interacting ionization and HG channels, and to be sensitive to the bound state wave function's symmetry.

Large-angle electron diffraction structure in laser-induced rescattering from rare gases

We have measured full momentum images of electrons rescattered from Xe, Kr, and Ar following the liberation of the electrons from these atoms by short, intense laser pulses. At high momenta the spectra show angular structure (diffraction) which is very target dependent and in good agreement with calculated differential cross sections for the scattering of free electrons from the corresponding ionic cores.

Rescattering for extended atomic systems

Laser-driven rescattering of electrons is the basis of many strong-field phenomena in atoms and molecules. Here, we will show how this mechanism operates in extended atomic systems, giving rise to effective energy absorption. Rescattering from extended systems can also lead to energy loss, which in its extreme form results in nonlinear light-induced trapping. Intense-laser interaction with atomic clusters is discussed as an example. We explain fast electron emission, seen in experimental and numerically obtained spectra, by rescattering of electrons at the highly charged cluster.

Molecular recollision interferometry in high harmonic generation

We use extreme-ultraviolet interferometry to measure the phase of high-order harmonic generation from transiently aligned CO(2) molecules. We unambiguously observe a reversal in phase of the high-order harmonic emission for higher harmonic orders with a sufficient degree of alignment. This results from molecular-scale quantum interferences between the molecular electronic wave function and the recolliding electron as it recombines with the molecule, and is consistent with a two-center model. Furthermore, using the combined harmonic intensity and phase information, we extract accurate information on the dispersion relation of the returning electron wave packet as a function of harmonic order. This analysis shows evidence of the effect of the molecular potential on the recolliding electron wave.

Quantum path interference in the wavelength dependence of high-harmonic generation

We investigate the dependence of the intensity of radiation due to high-harmonic generation as a function of the wavelength lambda of the fundamental driver field. Superimposed on a smooth power-law dependence observed previously, we find surprisingly strong and rapid fluctuations on a fine lambda scale. We identify the origin of these fluctuations in terms of quantum path interferences with up to five returning orbits significantly contributing.

Polarization state of high-order harmonic emission from aligned molecules

High harmonic emission in isotropic gases is polarized in the same direction as the incident laser polarization. Laser-induced molecular alignment allows us to break the symmetry of the gas medium. By using aligned molecules in high harmonic generation experiments, we show that the polarization of the extreme ultraviolet emission depends strongly on the molecular alignment and the orbital structure. Polarization measurements give insight into the molecular orbital symmetry. Furthermore, molecular alignment will allow us to produce attosecond pulses with time-dependent polarization.

Effective fields in laser-driven electron recollision and charge localization

We propose a model to describe correlated two-electron dynamics in strong laser fields during laser-induced recollision between an electron and its parent ion. We derive an effective interaction potential which describes the effect of the laser-driven electron collision with an ion while retaining the correlation between the colliding and the bound electron. Using dissociative ionization of molecular hydrogen as an example, we analyze the dynamics of correlation-driven electron localization in a dissociating hydrogen molecular ion.

Information content of high harmonics generated from aligned molecules

We derive an expression for the harmonic signal from nonadiabatically aligned molecules that accounts for both electronic and rotational motions. We identify a single approximation, which converts the expression into a physically transparent and computationally convenient form. Our analytical result gives explicitly the time dependence of the harmonic spectra, thus explaining the observations of a class of recent experiments. Moreover, it points to new opportunities for generating insights into the structure and dynamics of molecular systems through harmonic generation experiments from aligned molecules. This includes information regarding the rotational and electronic dynamics of isolated systems, as well as regarding the decoherence and relaxation in molecules subject to a dissipative environment.

Harnessing attosecond science in the quest for coherent X-rays

Modern laser technology has revolutionized the sensitivity and precision of spectroscopy by providing coherent light in a spectrum spanning the infrared, visible, and ultraviolet wavelength regimes. However, the generation of shorter-wavelength coherent pulses in the x-ray region has proven much more challenging. The recent emergence of high harmonic generation techniques opens the door to this possibility. Here we review the new science that is enabled by an ability to manipulate and control electrons on attosecond time scales, ranging from new tabletop sources of coherent x-rays to an ability to follow complex electron dynamics in molecules and materials. We also explore the implications of these advances for the future of molecular structural characterization schemes that currently rely so heavily on scattering from incoherent x-ray sources.

Ultrafast electronuclear dynamics of H(2) double ionization

The ultrafast electronic and nuclear dynamics of H(2) laser-induced double ionization is studied using a time-dependent wave packet approach that goes beyond the fixed nuclei approximation. The double ionization pathways are analyzed by following the evolution of the total wave function during and after the pulse. The rescattering of the first ionized electron produces a coherent superposition of excited molecular states which presents a pronounced transient H(+)H(-) character. This attosecond excitation is followed by field-induced double ionization and by the formation of short-lived autoionizing states which decay via double ionization. These two double ionization mechanisms may be identified by their signatures imprinted in the kinetic-energy distribution of the ejected protons.

Probing orbital structure of polyatomic molecules by high-order harmonic generation

The effects of electronic structure and symmetry are observed in laser driven high-order harmonic generation for laser aligned conjugated polyatomic molecular systems. The dependence of the harmonic yield on the angle between the molecular axis and the polarization of the driving laser field is seen to contain the fingerprint of the highest occupied molecular orbitals in acetylene and allene, a good quantitative agreement with calculations employing the strong field approximation was found. These measurements support the extension of the recently proposed molecular orbital imaging techniques beyond simple diatomic molecules to larger molecular systems.

High harmonic generation and the role of atomic orbital wave functions

High harmonic spectra were recorded from different rare-gas atoms under identical experimental conditions. It is shown that although each atom's spectrum is different, the differences are due almost entirely to the orbital influence in the recombination step. The amplitude of the continuum electron wave packet versus kinetic energy is derived from these data and is shown to be largely independent of the atom, in agreement with models of tunnel ionization. We compare the measurements with calculations in both the length gauge and the velocity gauge and show that the two gauges imply a different de Broglie wavelength.

Destructive interference during high harmonic generation in mixed gases

We report on the first experimental evidence of the destructive and constructive interference of high harmonics generated in a mixed gas of He and Ne, which facilitates the coherent control of high harmonic generation. Theoretically, we develop an analytical model of high harmonic generation in mixed gases and succeed in reproducing the experimental results and deriving the optimization conditions for the process. The observed interference modulation is attributed to the difference between the phases of the intrinsically chirped harmonic pulses from He and Ne, which leads to a novel method for broadband measurement of the harmonic phases and for observing the underlying attosecond electron dynamics.

Origin of anomalous spectra of dynamic alignments observed in N2 and O2

Recent pump-probe experiments with intense femtosecond laser pulses and diatomic molecules N2 and O2, have revealed the presence of Raman-forbidden anomalous series and lines in the Fourier spectrum of HHG (high harmonic generation) signals. A theoretical analysis of the problem is made by deriving a general expression of the angle dependent HHG operator that governs the dynamic alignment signals in linear molecules, and applying them to the experiments in N2 and O2. A unified interpretation of the origin of the observed Raman-allowed and the anomalous spectral features is given. The results are also used to estimate the molecular temperature in the experiments.

Direct XUV probing of attosecond electron recollision

We demonstrate that the recolliding electron wave packet, fundamental to many strong field phenomena, can be directly imaged with sub-A spatial and attosecond temporal resolution using attosecond extreme ultraviolet (XUV) pulses. When the recolliding electron revisits the parent ion, it can absorb an XUV photon yielding high energy electron and thereby providing a measurement of the electron energy at the moment of recollision. The full temporal evolution of the recollision wave packet can be reconstructed by measuring the photoelectron spectra for different time delays between the driving laser and the attosecond XUV probe. The strength of the photoelectron signal can be used to characterize the spatial distribution of the electron density in the longitudinal direction. Elliptical polarization can be used to characterize the electron probability in transversal direction.

Probe of high-order harmonic generation in a hollow waveguide geometry using counterpropagating light

We use counterpropagating light to directly observe the coherent buildup of high harmonic generation in a hollow waveguide geometry. We measure, for the first time, coherence lengths for high photon energies that cannot be phase matched using conventional approaches. We also probe the transition through phase matching, the ionization level at which different harmonic orders are generated, and the change in the coherence length as the driving laser is depleted. These results directly prescribe the optimal structures or pulse trains required for implementing quasiphase matching.

High harmonic generation and molecular orbital tomography in multielectron systems

High harmonic radiation is produced when atoms or molecules are ionized by an intense femtosecond laser pulse. The radiated spectrum has been shown experimentally to contain information on the electronic structure of the molecule, which can be interpreted as an image of a single molecular orbital. Previous theory for high harmonic generation has been limited to the single-active-electron approximation. Utilizing semisudden approximation, the authors develop a theory of the recombination step in high harmonic generation and tomographic reconstruction in multielectron systems, taking into account electron spin statistics and electron-electron correlations within the parent molecule and the ion. They show that the resulting corrections significantly modify the theoretical predictions, and bring them in a better agreement with experiment. They further show that exchange contributions to harmonic radiation can be used to extract additional information on the electronic wave function.

Transient phase masks in high-harmonic generation

We present a method for controlling the spatial properties of high-harmonic beams with high efficiency. The high nonlinearity of harmonic generation allows weak control beams to induce a phase mask for the extreme UV light as it is formed. We fabricate a phase grating and demonstrate efficient diffraction in the far field. Diffractive elements formed in this way are transient. Since they are induced by the subcycle interaction of the medium with the fundamental and control fields, they can be extended to the attosecond time scale.

Ellipticity dependence of high-order harmonic generation from aligned molecules

We report ellipticity dependence of high-order harmonic generation (HHG) from aligned N2, O2, and CO2 molecules. Experimentally, we find that the ellipticity dependence is sensitive to molecular alignment and to the shape and symmetry of the valence orbitals. It is also found that the destructive interference in the recombination process affects the ellipticity dependence. Theoretically, we extend the original Lewenstein model to a more generalized model, which can be applicable to HHG from molecules, by introducing an electron acceleration parameter xi(theta) and by combining the molecular orbital method. The present observations are successfully explained by our model.

Unified microscopic-macroscopic formulation of high-order difference-frequency mixing in plasmas

We investigate high-order difference-frequency mixing in plasmas, taking into account the microscopic rescattering physics and propagation effects for the first time. We show that phase matching can occur over a broad frequency range, up to very high photon energies, and that it is confined to specific temporal and spatial windows. This gated phase matching mechanism is driven by the continuous phase slip between two driving fields and can be employed for manipulating the temporal, spatial, and spectral properties of high harmonic emission.

Nonsequential double recombination in intense laser fields

A second plateau in the harmonic spectra of laser-driven two-electron atoms is observed both in the numerical solution of a low-dimensional model helium atom and using an extended strong field approximation. It is shown that the harmonics well beyond the usual cutoff are due to the simultaneous recombination of the two electrons, which were emitted during different, previous half-cycles. The new cutoff is explained in terms of classical trajectories. Classical predictions and the time-frequency analysis of the ab initio quantum results are in excellent agreement. The mechanism corresponds to the inverse single photon double ionization process in the presence of a (low frequency) laser field.

Efficient isolated attosecond pulse generation from a multi-cycle two-color laser field

We present a new scheme to generate efficient isolated attosecond pulse by using the combination of a fundamental and a weak second harmonic (SH) field in the multi-cycle regime. Because of the symmetry breaking of the electric field, the ionization dynamics of the electron can be controlled by adjusting the relative phase of the two fields. Then attosecond (as) pulses will be generated each full optical cycle of the fundamental field. Our simulation shows that the intensity of the single attosecond pulse can be enhanced by an order of magnitude in our scheme compared with the scheme in [Opt. Lett. 31, 975 (2006)].

Scaling of wave-packet dynamics in an intense midinfrared field

A theoretical investigation is presented that examines the wavelength scaling from near-visible (0.8 micro m) to midinfrared (2 micro m) of the photoelectron distribution and high harmonics generated by a "single" atom in an intense electromagnetic field. The calculations use a numerical solution of the time-dependent Schrödinger equation (TDSE) in argon and the strong-field approximation in helium. The scaling of electron energies (lambda2), harmonic cutoff (lambda2), and attochirp (lambda -1) agree with classical mechanics, but it is found that, surprisingly, the harmonic yield follows a lambda -(5-6) scaling at constant intensity. In addition, the TDSE results reveal an unexpected contribution from higher-order returns of the rescattering electron wave packet.

Laser phase control of high-order harmonic generation at large internuclear distance: the H+ -H2+ system

Exact (Born-Oppenheimer) 3-D numerical solutions of the time-dependent Schrödinger equation are obtained for the one electron linear H+-H2+ atom-molecule system at large internuclear distance R in interaction with two-cycles intense (I>10(14) W cm(-2)) 800 nm laser pulses. High-order harmonic generation (HHG) spectra are obtained with an energy cutoff larger than the atomic maximum of I(p)+3U(p), where I(p) is the ionization potential and U(p) is the ponderomotive energy. At large R, this extended cutoff is shown to be related to the nature of electron transfer, whose direction is shown to depend critically on the carrier-envelope phase (CEP) of the ultrashort pulse. Constructive and destructive interferences in the HHG spectrum resulting from coherent superpositions of electronic states in the H+-H2+ system are interpreted in terms of multiple electron trajectories extracted from a time profile analysis.

Single attosecond pulses from high harmonics driven by self-compressed filaments

We show that isolated subfemtosecond, extreme ultraviolet (XUV) pulses can be generated via harmonic generation in argon by few-cycle infrared pulses formed through filamentation-induced self-compression in neon. Our calculations show that the time structure of the XUV pulses depends sensitively on both the amplitude and the phase modulation that are induced in the driving pulse during the self-compression process.

Generating single attosecond pulses via spatial filtering

The first observation of isolated attosecond pulses by Hentschel [Nature 414, 509 (2001)] resulted from an experiment that left the exact mechanism of their generation unresolved. A complete simulation of the experiment reveals the reason for its success: single pulses were efficiently isolated from two or more generated pulses by spatial filtering in the far field. Our explanation suggests a new, simple paradigm for the production of isolated attosecond bursts. We show that this method can be used, in conjunction with carrier-envelope phase stabilization, to select single attosecond pulses by use of 10 fs driving pulses.

Isolated Single-Cycle Attosecond Pulses

We generated single-cycle isolated attosecond pulses around approximately 36 electron volts using phase-stabilized 5-femtosecond driving pulses with a modulated polarization state. Using a complete temporal characterization technique, we demonstrated the compression of the generated pulses for as low as 130 attoseconds, corresponding to less than 1.2 optical cycles. Numerical simulations of the generation process show that the carrier-envelope phase of the attosecond pulses is stable. The availability of single-cycle isolated attosecond pulses opens the way to a new regime in ultrafast physics, in which the strong-field electron dynamics in atoms and molecules is driven by the electric field of the attosecond pulses rather than by their intensity profile.

Interferometric autocorrelation of an attosecond pulse train in the single-cycle regime

We report on the direct observation of the phase locking of the attosecond pulse train (APT) via interferometric autocorrelation in the extreme ultraviolet region. APT is formed with Fourier synthesis of high-order harmonic fields of a femtosecond laser pulse. Time-of-flight mass spectra of N+, resulting from the Coulomb explosion of N2 absorbing two photons of APT, efficiently yield correlated signals of APT. The measured autocorrelation trace exhibits that the duration of the pulse should be only 1.3 periods of the extreme ultraviolet carrier frequency. A few interference fringes within the short pulse duration clearly show two types of symmetry, which ensure the phase locking between pulses in APT.

Enhanced recollisions for antisymmetric molecular orbitals in intense laser fields

The peculiarities of antisymmetric molecular orbitals are investigated in very intense linearly polarized laser pulses. For this purpose, the ionization-recollision quantum dynamics is evaluated theoretically beyond the dipole approximation. As opposed to the usual situation, the laser magnetic field component is found to strongly enhance recollision probabilities for particularly oriented antisymmetric molecular orbitals. Harmonic generation and related processes are thus allowed at high laser intensities without the common limitations by the laser magnetic field.

High harmonic generation and molecular orbital tomography in multielectron systems: beyond the single active electron approximation

It was recently shown that the highest molecular orbital of N2 could be reconstructed from a series of high harmonic measurements. Existing theories of high harmonic generation and orbital tomographic imaging are based on the single active electron approximation that ignores essential quantum mechanical properties such as the indistinguishability of identical particles and the Pauli exclusion principle. We show that, when fully antisymmetrized multielectron wave functions and electronic relaxation in the cation are considered, molecular orbital tomography records the image of the Dyson orbital plus exchange contributions from inner shells. The mixing of contributions from more than one molecular orbital gives access to additional wave function information. By utilizing the exchange term, harmonic emission from a closed-shell 4-electron system can be interpreted as a complete Hartree-Fock wave function.

Monitoring molecular dynamics using coherent electrons from high harmonic generation

We report a previously undescribed spectroscopic probe that makes use of electrons rescattered during the process of high-order harmonic generation. We excite coherent vibrations in SF(6) using impulsive stimulated Raman scattering with a short laser pulse. A second, more intense laser pulse generates high-order harmonics of the fundamental laser, at wavelengths of approximately 20-50 nm. The high-order harmonic yield is observed to oscillate, at frequencies corresponding to all of the Raman-active modes of SF(6), with an asymmetric mode most visible. The data also show evidence of relaxation dynamics after impulsive excitation of the molecule. Theoretical modeling indicates that the high harmonic yield should be modulated by both Raman and infrared-active vibrational modes. Our results indicate that high harmonic generation is a very sensitive probe of vibrational dynamics and may yield more information simultaneously than conventional ultrafast spectroscopic techniques. Because the de Broglie wavelength of the recolliding electron is on the order of interatomic distances, i.e., approximately 1.5 A, small changes in the shape of the molecule lead to large changes in the high harmonic yield. This work therefore demonstrates a previously undescribed spectroscopic technique for probing ultrafast internal dynamics in molecules and, in particular, on the chemically important ground-state potential surface.